Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 914 · 40 questions

How to solve it (Patterns repeat): Divisibility: 3 and 9 — digit sum; 4 — last two digits; 8 — last three digits; 6 — even and divisible by 3; 11 — alternate digit sums differ by 0 or a multiple of 11. Remainder: find the biggest multiple not more than the number and take it away; to make a number divisible, add divisor − remainder. Units digit of a power: write the cycle (7, 9, 3, 1 for 7), divide the power by the cycle length and use the remainder.

1. 1.What is the remainder when 347 is divided by 9? (a) 7 (b) 6 (c) 4 (d) 5
2. 2.Which of these numbers is divisible by 6? (a) 2126 (b) 2116 (c) 2119 (d) 2118
3. 3.What is the smallest number you must add to 179 to make it divisible by 9? (a) 1 (b) 2 (c) 8 (d) 3
4. 4.Which of these numbers is divisible by 11? (a) 8437 (b) 8439 (c) 8456 (d) 8428
5. 5.What is the remainder when 99 is divided by 8? (a) 4 (b) 3 (c) 2 (d) 5
6. 6.What is the smallest number you must add to 146 to make it divisible by 4? (a) 3 (b) 1 (c) 2
7. 7.What is the remainder when 224 is divided by 3? (a) 0 (b) 2 (c) 1
8. 8.Which of these numbers is divisible by 3? (a) 2549 (b) 2546 (c) 2544 (d) 2543
9. 9.What is the smallest number you must add to 273 to make it divisible by 4? (a) 2 (b) 1 (c) 3
10. 10.Which of these numbers is divisible by 3? (a) 200 (b) 196 (c) 202 (d) 198
11. 11.Which of these numbers is divisible by 8? (a) 3223 (b) 3229 (c) 3214 (d) 3216
12. 12.Which of these numbers is divisible by 6? (a) 2220 (b) 2228 (c) 2217 (d) 2221
13. 13.What is the units digit of 4^59? (a) 5 (b) 3 (c) 4 (d) 6
14. 14.What is the remainder when 314 is divided by 12? (a) 1 (b) 3 (c) 10 (d) 2
15. 15.Which of these numbers is divisible by 8? (a) 2200 (b) 2202 (c) 2199 (d) 2210
16. 16.What is the units digit of 9^30? (a) 9 (b) 2 (c) 0 (d) 1
17. 17.What is the smallest number you must add to 518 to make it divisible by 10? (a) 8 (b) 2 (c) 1 (d) 3
18. 18.What is the remainder when 59 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
19. 19.What is the smallest number you must add to 105 to make it divisible by 4? (a) 2 (b) 1 (c) 3
20. 20.What is the remainder when 410 is divided by 7? (a) 5 (b) 3 (c) 4 (d) 6
21. 21.What is the units digit of 4^20? (a) 4 (b) 7 (c) 6 (d) 5
22. 22.What is the remainder when 226 is divided by 3? (a) 1 (b) 2 (c) 0
23. 23.What is the smallest number you must add to 212 to make it divisible by 11? (a) 8 (b) 7 (c) 9 (d) 3
24. 24.What is the units digit of 2^45? (a) 8 (b) 4 (c) 2 (d) 6
25. 25.What is the smallest number you must add to 364 to make it divisible by 6? (a) 4 (b) 1 (c) 3 (d) 2
26. 26.Which of these numbers is divisible by 6? (a) 4347 (b) 4339 (c) 4354 (d) 4344
27. 27.What is the smallest number you must add to 178 to make it divisible by 3? (a) 1 (b) 2
28. 28.What is the remainder when 198 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
29. 29.What is the smallest number you must add to 109 to make it divisible by 4? (a) 2 (b) 3 (c) 1
30. 30.What is the remainder when 218 is divided by 11? (a) 2 (b) 10 (c) 9 (d) 8
31. 31.What is the units digit of 2^60? (a) 2 (b) 6 (c) 4 (d) 8
32. 32.What is the smallest number you must add to 232 to make it divisible by 3? (a) 2 (b) 1
33. 33.What is the remainder when 674 is divided by 9? (a) 1 (b) 6 (c) 7 (d) 8
34. 34.Which of these numbers is divisible by 6? (a) 685 (b) 690 (c) 697 (d) 692
35. 35.Which of these numbers is divisible by 8? (a) 2064 (b) 2070 (c) 2061 (d) 2073
36. 36.Which of these numbers is divisible by 11? (a) 4521 (b) 4524 (c) 4517 (d) 4542
37. 37.What is the units digit of 3^30? (a) 1 (b) 3 (c) 9 (d) 7
38. 38.What is the remainder when 95 is divided by 3? (a) 0 (b) 1 (c) 2
39. 39.What is the units digit of 2^46? (a) 8 (b) 4 (c) 6 (d) 2
40. 40.What is the smallest number you must add to 139 to make it divisible by 4? (a) 1 (b) 2 (c) 3

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